Bounded-$\sigma$ criterion for hereditary closures of linked chain graphs

Let $(G_n)_{n \geq 1}$ be a family of linked chain graphs with linking permutations $\pi_1,\pi_2,\dots$, and let $\mathcal X$ be the hereditary closure of this family. Write $G_{\pi_i}$ for the permutation graph associated with $\pi_i$. Linked-chain criterion conjecture. The parameter $\sigma$ is bounded in $\mathcal X$ if and only if the hereditary closure of the permutation graphs $G_{\pi_i}$ does not contain all unions of cliques or all complete bipartite graphs. The conjecture is proposed as a characterization in the linked-chain setting; the paper does not establish either direction in general.